On volumes and filling collections of multicurves

Let $S$ be a surface of negative Euler characteristic and consider a finite filling collection $Γ$ of closed curves on $S$ in minimal position. An observation of Foulon and Hasselblatt shows that $PT(S) \setminus \hatΓ$ is a finite-volume hyperbolic 3-manifold, where $PT(S)$ is the projectivized tangent bundle and $\hatΓ$ is the set of tangent lines to $Γ$. In particular, $vol(PT(S) \setminus \hatΓ)$ is a mapping class group invariant of the collection $Γ$. When $Γ$ is a filling pair of simple closed curves, we show that this volume is coarsely comparable to Weil-Petersson distance between strata in Teichmüller space. Our main tool is the study of stratified hyperbolic links $\barΓ$ in a Seifert-fibered space $N$ over $S$. For such links, the volume of $N\setminus\barΓ$ is coarsely comparable to expressions involving distances in the pants graph.

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